Regular homotopy

Regular homotopy

In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. In particular, the homotopy must go through immersions and extend continuously to a homotopy of the tangent bundle.

Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them.

The Whitney-Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if their Gauss maps have the same winding number. Stephen Smale classified the regular homotopy classes of an "k"-sphere immersed in mathbb R^n. A corollary of his work is that there is only one regular homotopy class of a "2"-sphere immersed in mathbb R^3. In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".

References

*Hassler Whitney, " [http://www.numdam.org/numdam-bin/item?id=CM_1937__4__276_0 On regular closed curves in the plane] ". Compositio Mathematica, 4 (1937), p. 276-284
*Stephen Smale, "A classification of immersions of the two-sphere". Trans. Amer. Math. Soc. 90 1958 281--290.
*Stephen Smale, "The classification of immersions of spheres in Euclidean spaces". Ann. of Math. (2) 69 1959 327--344.


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